Isotropic random fields with infinitely divisible marginal distributions

Abstract

A simple but efficient approach is proposed in this paper to construct the isotropic random field in (d 2), whose univariate marginal distributions may be taken as any infinitely divisible distribution with finite variance. The three building blocks in our building tool box are a second-order Levy process on the real line, a d-variate random vector uniformly distributed on the unit sphere, and a positive random variable that generates a Polya-type function. The approach extends readily to the multivariate case and results in a vector random field in with isotropic direct covariance functions and with any specified infinitely divisible marginal distributions. A characterization of the turning bands simulation feature is also derived for the covariance matrix function of a Gaussian or elliptically contoured random field that is isotropic and mean square continuous in R-d.